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Abstract

This paper is concerned with the existence of solutions to a class of p-Kirchhoff
type equations with Neumann boundary data as follows:

By means of a direct variational approach, we establish conditions ensuring the existence
and multiplicity of solutions for the problem.

Keywords:

Nonlocal problems; Neumann problem; p-Kirchhoff's equation

1. Introduction

In this paper, we deal with the nonlocal p-Kirchhoff type of problem given by:

(1.1)

where Ω is a smooth bounded domain in RN, 1 < p < N, ν is the unit exterior vector on ∂Ω, Δp is the p-Laplacian operator, that is, Δpu = div(|∇u|p−2∇u), the function M : R+ → R+ is a continuous function and there is a constant m0 > 0, such that

is a continuous function and satisfies the subcritical condition:

(1.2)

where C denotes a generic positive constant.

Problem (1.1) is called nonlocal because of the presence of the term M, which implies that the equation is no longer a pointwise identity. This provokes
some mathematical difficulties which makes the study of such a problem particulary
interesting. This problem has a physical motivation when p = 2. In this case, the operator M(∫Ω|∇u|2dx)Δu appears in the Kirchhoff equation which arises in nonlinear vibrations, namely

P-Kirchhoff problem began to attract the attention of several researchers mainly after
the work of Lions [1], where a functional analysis approach was proposed to attack it. The reader may consult
[2-8] and the references therein for similar problem in several cases.

This work is organized as follows, in Section 2, we present some preliminary results
and in Section 3 we prove the main results.

2. Preliminaries

By a weak solution of (1.1), then we say that a function u ε W1,p(Ω) such that

So we work essentially in the space W1,p(Ω) endowed with the norm

and the space W1,p(Ω) may be split in the following way. Let Wc = 〈1〉, that is, the subspace of W1,p(Ω) spanned by the constant function 1, and , which is called the space of functions of W1,p(Ω) with null mean in Ω. Thus

As it is well known the Poincaré's inequality does not hold in the space W1,p(Ω). However, it is true in W0.

Lemma 2.1 [8] (Poincaré-Wirtinger's inequality) There exists a constant η > 0 such that for all z ∈ W0.

Let us also recall the following useful notion from nonlinear operator theory. If
X is a Banach space and A : X → X* is an operator, we say that A is of type (S+), if for every sequence {xn}n≥1 ⊆ X such that xn ⇀ x weakly in X, and . we have that xn → x in X.

Let us consider the map A : W1,p(Ω) → W1,p(Ω)* corresponding to −Δp with Neumann boundary data, defined by

Definition 2.1. Let E be a real Banach space, and D an open subset of E. Suppose that a functional
J : D → R is Fréchet differentiable on D. If x0 ∈ D and the Fréchet derivative J' (x0) = 0, then we call that x0 is a critical point of the functional J and c = J(x0) is a critical value of J.

Definition 2.2. For J ∈ C1(E, R), we say J satisfies the Palais-Smale condition (denoted by (PS)) if any sequence {un} ⊂ E for which J(un) is bounded and J'(un) → 0 as n → ∞ possesses a convergent subsequence.

In the present paper, we give an existence theorem and a multiplicity theorem for
problem (1.1). Our main results are the following two theorems.

Theorem 2.1 If following hold:

(F0) , where , η appears in Lemma 2.1;

(F1) ;

(F2).

Then the problem (1.1) has least three distinct weak solutions in W1,p(Ω).

Theorem 2.2 If the following hold:

(M1) The function M that appears in the classical Kirchhoff equation satisfies for all t ≥ 0, where ;

(F3);

(F4);

(F5)
.

Then the problem (1.1) has at least one weak solution in W1,p(Ω).

Remark 2.2 We exhibit now two examples of nonlinearities that fulfill all of our hypotheses

hypotheses (F0), (F1), (F2) and (1.2) are clearly satisfied.

hypotheses (F3), (F4) and (F5) and (1.2) are clearly satisfied.

3. Proofs of the theorems

Let us start by considering the functional J : W1,p(Ω) → R given by

Proof of Theorem 2.1 By (F0), we know that f(x, 0) = 0, and hence u(x) = 0 is a solution of (1.1).

To complete the proof we prove the following lemmas.

Lemma 3.1 Any bounded (PS) sequence of J has a strongly convergent subsequence.

Proof: Let {un} be a bounded (PS) sequence of J. Passing to a subsequence if necessary, there exists u ∈ W1,p(Ω) such that un ⇀ u. From the subcritical growth of f and the Sobolev embedding, we see that

and since J'(un)(un − u) → 0, we conclude that

In view of condition (M0), we have

Using Lemma 2.2, we have un → u as n → ∞. □

Lemma 3.2 If condition (M0), (F1) and (F2) hold, then .

Proof: If there are a sequence {un} and a constant C such that ||un|| → ∞ as n → ∞, and J(un) ≤ C (n = 1, 2 ···), let , then there exist v0 ∈ W1,p(Ω) and a subsequence of {vn}, we still note by {vn}, such that vn ⇀ v0 in W1,p(Ω) and vn → v0 in Lp(Ω).

For any ε > 0, by (F1), there is a H > 0 such that for all |u| ≥ H and a.e. x ∈ Ω, then there exists a constant C > 0 such that for all u ∈ R, and a.e. x ∈ Ω, Consequently

It implies ∫Ω|v0|pdx ≥ 1. On the other hand, by the weak lower semi-continuity of the norm, one has

Hence , so |v0(x)| = constant ≠ 0 a.e. x ∈ Ω. By (F2), . Hence

This is a contradiction. Hence J is coercive on W1,p(Ω), bounded from below, and satisfies the (PS) condition. □

By Lemma 3.1 and 3.2, we know that J is coercive on W1,p(Ω), bounded from below, and satisfies the (PS) condition. From condition (F0), we know, there exist r > 0, ε > 0 such that

If inf{J(u), u ∈ W1,p(Ω)} = 0, then all u ∈ Wc with ||u|| ≤ ρ are minimum of J, which implies that J has infinite critical points. If inf{J(u), u ∈ W1,p(Ω)} < 0 then by Lemma 2.3, J has at least two nontrivial critical points. Hence problem (1.1) has at least two
nontrivial solutions in W1,p(Ω), Therefore, problem (1.1) has at least three distinct solutions in W1,p(Ω). □

Proof of Theorem 2.2. We divide the proof into several lemmas.

Lemma 3.3 If condition (F3) and (F5) hold, then is anticoercive. (i.e. we have that J(u) → -∞, as |u| → ∞, u ∈ R.)

Proof: By virtue of hypothesis (F5), for any given L > 0, we can find R1 = R1(L) > 0 such that

Thus, using hypothesis (F3), we have

So

Since L > 0 is arbitrary, it follows that

and so

This proves that is anticoercive. □

Lemma 3.4 If hypothesis (F4) holds, then
.

Proof: For a given , we can find Cε > 0 such that for a.e. x ∈ Ω all u ∈ R. Then

We claim that the sequence {un} is bounded. We argue by contradiction. Suppose that ||u|| → +∞, as n → ∞, we set , ∀n ≥ 1. Then ||vn|| = 1 for all n ≥ 1 and so, passing to a subsequence if necessary, we may assume that

from (3.2), we have ∀h ∈ W1,p(Ω)

(3.3)

with εn ↓ 0.

In (3.3), we choose h = vn − v ∈ W1,p(Ω), note that by virtue of hypothesis (F4), we have

where .

So we have

Since M(t) > m0 for all t ≥ 0, so we have

Hence, using the (S+) property, we have vn → v in W1,p(Ω) with ||v|| = 1, then v ≠ 0. Now passing to the limit as n → ∞ in (3.3), we obtain